The results of this paper show that the structure of sets mentioned in the title is not trivial. For example, it is shown that there exist countable sets of uniqueness for logarithmic potential, i.e., closed countable subsets $E$ of the unit circle $\mathbb T$ such that $$ f\in C(\mathbb T),\ f\mid_E=0,\ U^f\mid_E=0\ \Rightarrow f\equiv0. $$ Here $U^f(z)=\frac1\pi\int_0^{2\pi}f(e^{i\theta})\log\frac1{|z-e^{i\theta}|}\,d\theta$. On the other hand, it is shoum that every countable porous closed subset of $\mathbb T$ is a nonuniqueness set. Bibliography: 9 titles.